A molecular vibration occurs when atoms The atom is a basic unit of matter that consists of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons . The electrons of an atom are bound to the nucleus by the electromagnetic force. Likewise, a group of atoms can remain in a molecule A molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense. In organic chemistry and biochemistry, the term molecule is used less strictly and also is applied to charged organic molecules are in periodic motion In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency. A nonlinear molecule with n atoms has 3n−6 normal modes of vibration, whereas a linear molecule has 3n−5 normal modes of vibration because rotation about its molecular axis is simply a rotation of the reference frame and cannot be observed[1]. A diatomic molecule Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen, nitrogen, oxygen, and carbon monoxide. Most elements aside from the noble gases form diatomic molecules when heated, but high temperatures—sometimes thousands thus has only one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other, each involving simultaneous vibrations of different parts of the molecule.

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E=hν, where h is Planck's constant. A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. The ground state of a quantum field theory is usually called the vacuum state or the vacuum. When two quanta are absorbed the first overtone An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the acoustic behavior of musical instruments. Due to a is excited, and so on to higher overtones.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion In physics, simple harmonic motion is the motion of a simple harmonic oscillator, a periodic motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic Anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in simple harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large then other and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, as the potential energy of the molecule is more like a Morse potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy Infrared spectroscopy is the subset of spectroscopy that deals with the infrared region of the electromagnetic spectrum. It covers a range of techniques, the most common being a form of absorption spectroscopy. As with all spectroscopic techniques, it can be used to identify compounds or investigate sample composition. Infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy Raman spectroscopy is a spectroscopic technique used to study vibrational, rotational, and other low-frequency modes in a system. It relies on inelastic scattering, or Raman scattering, of monochromatic light, usually from a laser in the visible, near infrared, or near ultraviolet range. The laser light interacts with phonons or other excitations, which typically uses visible light, can also be used to measure vibration frequencies directly.

Vibrational excitation can occur in conjunction with electronic excitation (vibronic transition), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Contents

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene Ethylene is a gaseous organic compound with the formula C2H4. It is the simplest alkene (older name: olefin from its oil-forming property). Because it contains a carbon-carbon double bond, ethylene is classified as an unsaturated hydrocarbon. Ethylene is widely used in industry and also has a role in biology as a hormone. Ethylene is the most,

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH2 rocking, 2 CH2 wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

See infrared spectroscopy Infrared spectroscopy is the subset of spectroscopy that deals with the infrared region of the electromagnetic spectrum. It covers a range of techniques, the most common being a form of absorption spectroscopy. As with all spectroscopic techniques, it can be used to identify compounds or investigate sample composition. Infrared spectroscopy for some animated illustrations of internal coordinates.

Symmetry-adapted coordinates

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates.[2] The projection operator is constructed with the aid of the character table of the molecular point group In chemistry, a point group is a group of geometric symmetries leaving a point fixed. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by

Qs1 = q1 + q2 + q3 + q4
Qs2 = q1 + q2 - q3 - q4
Qs3 = q1 - q2 + q3 - q4
Qs4 = q1 - q2 - q3 + q4

where q1 - q4 are the internal coordinates for stretching of each of the four C-H bonds.

Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto.[3]

Normal coordinates

A normal coordinate, Q, may sometimes be constructed directly as a symmetry-adapted coordinate. This is possible when the normal coordinate belongs uniquely to a particular irreducible representation In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they of the molecular point group In chemistry, a point group is a group of geometric symmetries leaving a point fixed. For example, the symmetry-adapted coordinates for bond-stretching of the linear carbon dioxide Carbon dioxide is a chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom. It is a gas at standard temperature and pressure and exists in Earth's atmosphere in this state. CO2 is a trace gas comprising 0.039% of the atmosphere molecule, O=C=O are both normal coordinates:

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide Hydrogen cyanide is a chemical compound with chemical formula HCN. Hydrogen cyanide is a colorless, extremely poisonous liquid that boils slightly above room temperature at 26 °C (79 °F). Hydrogen cyanide is a linear molecule, with a triple bond between carbon and nitrogen. A minor tautomer of HCN is HNC, hydrogen isocyanide, HCN, The two stretching vibrations are

  1. principally C-H stretching with a little C-N stretching; Q1 = q1 + a q2 (a << 1)
  2. principally C-N stretching with a little C-H stretching; Q2 = b q1 + q2 (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.[4]

Newtonian mechanics

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics, to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hooke's: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere.[5]

By Newton’s second law of motion Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several different ways over nearly three centuries, and can be summarised as follows: this force is also equal to a "mass", m, times acceleration.

Since this is one and the same force the ordinary differential equation In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable follows.

The solution to this equation of simple harmonic motion In physics, simple harmonic motion is the motion of a simple harmonic oscillator, a periodic motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction is

A is the maximum amplitude of the vibration coordinate Q. It remains to define the "mass", m. In a homonuclear diatomic molecule such as N2, it is half the mass of one atom. In a heteronuclear diatomic molecule, AB, it is the reduced mass Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not reduced. In the computation one mass can, μ given by

The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,νi are obtained from the eigenvalues In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. The prefix eigen is the German word for innate, distinct. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are,λi, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.[4] F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in [6].

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

,

where n is a quantum number that can take values of 0, 1, 2 ... The difference in energy when n changes by 1 are therefore equal to the energy derived using classical mechanics. See quantum harmonic oscillator The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical for graphs of the first 5 wave functions. Knowing the wave functions, certain selection rules can be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band.

Intensities

In an infrared spectrum the intensity In physics, intensity is a measure of the energy flux, averaged over a certain time period. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech. For example, "the intensity of pressure" is meaningless, since the parameters of of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate.[7] The intensity of Raman bands depends on polarizability Polarizability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field, which may be caused by the presence of a nearby ion or dipole.

See also

References

  1. ^ Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover)
  2. ^ F.A. Cotton Chemical applications of group theory, Wiley, 1962, 1971
  3. ^ K. Nakamoto Infrared and Raman spectra of inorganic and coordination compounds, 5th. edition, Part A, Wiley, 1997
  4. ^ a b E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations, McGraw-Hill, 1955. (Reprinted by Dover 1980)
  5. ^ S. Califano, Vibrational states, Wiley, 1976
  6. ^ P. Gans, Vibrating molecules, Chapman and Hall, 1971
  7. ^ D. Steele, Theory of vibrational spectroscopy, W.B. Saunders, 1971

Further reading

External links

Categories: Chemical physics | Spectroscopy Spectroscopy is the study of the interaction between radiation and matter. Spectrometry is the measurement of these interactions and a machine which performs such measurements is a spectrometer or spectrograph. A plot of the interaction is referred to as a spectrogram, or, informally, a spectrum | Molecular vibration

 

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